The various surface lots near the Riverwalk are not grossly overpriced. Check the Riverwalk maps, pick one and park, somewhere in the middle, allowing strolling in both directions.
Be prepared for it to be "HOT". Your posting location is not given, but San Antonio Summers are severe.
Easy side trip downtown....The Mercado, a cheap aparking garage (on top), "souvenirs" and the always welcome break at "Mi Tierra", certainly not the best TexMex, but all day and all night, every day of the year, nothing like it.